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Tail probabilities and partial moments for quadratic forms in multivariate generalized hyperbolic random vectors

Simon Broda

No 13-04, UvA-Econometrics Working Papers from Universiteit van Amsterdam, Dept. of Econometrics

Abstract: Countless test statistics can be written as quadratic forms in certain random vectors, or ratios thereof. Consequently, their distribution has received considerable attention in the literature. Except for a few special cases, no closed-form expression for the cdf exists, and one resorts to numerical methods. Traditionally the problem is analyzed under the assumption of joint Gaussianity; the algorithm that is usually employed is that of Imhof (1961). The present manuscript generalizes this result to the case of multivariate generalized hyperbolic (MGHyp) random vectors. The MGHyp is a very flexible distribution which nests, among others, the multivariate t, Laplace, and variance gamma distributions. An expression for the first partial moment is also obtained, which plays a vital role in financial risk management. The proof involves a generalization of the classic inversion formula due to Gil-Pelaez (1951). Two applications are considered: first, the finite-sample distribution of the 2SLS estimator of a structural parameter. Second, the Value at Risk and Expected Shortfall of a quadratic portfolio with heavy-tailed risk factors.

Date: 2013-05-01
New Economics Papers: this item is included in nep-ecm and nep-rmg
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Working Paper: Tail Probabilities and Partial Moments for Quadratic Forms in Multivariate Generalized Hyperbolic Random Vectors (2013) Downloads
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